Consider relationships among the aforementioned graph classes.
Fan-crossing
and fan-crossing free are complementary on edges that are crossed at
least twice [1]}. Both admit edges that are uncrossed
or are crossed once. Hence, every 1-planar graph is both
fan-crossing and fan-crossing free, but not conversely, since there
are graphs that are fan-crossing and fan-crossing free, but not
1-planar [2]}. There are fan-crossing graphs that are
not fan-crossing free, since the density of fan-crossing free graphs
is \(4n-8\)  [3]}, whereas there are \(n\) -vertex
fan-crossing graphs with \(5n-10\)  edges [4]}, [5]}.
Particular examples for fan-crossing and non-fan-crossing
free graphs are \(K_7\)  and \(K_{p,q}\)  with \(p=3,4\)  and \(q \ge 7\) 
[6]}. We prove the opposite direction, which was
open so far [7]}. In particular, we show that
the 2-subdivision of \(K_n\)  for \(n \ge 12\)  and \(d\) -dimensional
cube-connected cycles \(CCC^{d}\)  for \(d \ge 11\)  are fan-crossing
free but neither fan-crossing nor 1-gap-planar. In addition, for
every \(k \ge 0\) , there are graphs that are RAC, and thus
fan-crossing free, but not \(k\) -gap-planar, and thus not \(2k\) -planar.
